Do you know how many Anytime seats are available (search your itinerary as a new booking, looking for one seat, then keep increasing the number, till the space shows unavailable). If there only are one or two open seats, you may have a more difficult time. At any rate, I suggest contacting Southwest via social media, sending a DM, noting your confirmation number, full name, and DOB to verify your identity. Tell them what date/flight number you prefer. Do not even mention that you checked availability. The social media team belongs to Customer Relations, so those CSRs have more ability to do overrides, as a customer accommodation.
As always, being polite and appreciative go a long way.
Has anyone had any luck being rebooked into a higher fare class at no change in fare? Yes...on a rebooking that Southwest really messed up with a silly inexcusable error.
Hope it works out. Please let us know if you had any luck.